A357415 -id:A357415 - cp's OEIS frontend

A339454 Number of subsets of {1..n} whose root mean square is an integer.

1, 2, 3, 4, 5, 6, 9, 10, 15, 20, 29, 52, 87, 166, 311, 538, 943, 1682, 2915, 5054, 8905, 15904, 28533, 51826, 95191, 175402, 325777, 607542, 1134191, 2128922, 3986433, 7485522, 14065135, 26446388, 49796025, 93920770, 177470237, 335780796, 636883269, 1209603646

Offset: 1

Ilya Gutkovskiy, Dec 05 2020

nonn

			a(9) = 15 subsets: {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {1, 7}, {1, 5, 7}, {1, 3, 5, 8, 9}, {3, 4, 5, 7, 9}, {1, 3, 5, 6, 8, 9} and {3, 4, 5, 6, 7, 9}.

Max Alekseyev, Table of n, a(n) for n = 1..100
Eric W. Weisstein's World of Mathematics, Root-Mean-Square

Cf. A051293, A240089, A326027, A339453.

from functools import lru_cache
from sympy.ntheory.primetest import is_square
def cond(sos, c): return c > 0 and sos%c == 0 and is_square(sos//c)
@lru_cache(maxsize=None)
def b(n, sos, c):
    if n == 0: return int(cond(sos, c))
    return b(n-1, sos, c) + b(n-1, sos+n*n, c+1)
a = lambda n: b(n, 0, 0)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Oct 06 2022

a(n) = A357415(n) + A357416(n). - Max Alekseyev, Mar 25 2025

a(23)-a(40) from Alois P. Heinz, Dec 05 2020

A357413 Number of nonempty subsets of {1..n} whose elements have an odd geometric mean.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 4, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 19, 19, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 39, 39, 40, 40, 49, 49, 50, 50, 51, 51, 52, 52, 53, 53, 54, 54, 55, 55, 62, 62, 63, 63, 64, 64, 65, 65, 66, 66, 67, 67, 90, 90, 91, 91, 92, 92

Offset: 0

Ilya Gutkovskiy, Sep 27 2022

nonn

The geometric mean of a subset such as in name must be an odd number in {1..n} which might ease the search for terms. - David A. Corneth, Sep 29 2022

			a(9) = 7 subsets: {1}, {3}, {5}, {7}, {9}, {1, 9} and {1, 3, 9}.

Max Alekseyev, Table of n, a(n) for n = 0..256

Cf. A001055, A326027, A357355, A357411, A357414, A357415.

from functools import lru_cache
from sympy import integer_nthroot
def cond(p, c): r, b = integer_nthroot(p, c); return b and r&1
@lru_cache(maxsize=None)
def b(n, p, c):
    if n == 0: return int (c > 0 and cond(p, c))
    return b(n-1, p, c) + b(n-1, p*n, c+1) if n&1 else b(n-1, p, c)
@lru_cache(maxsize=None)
def a(n): return b(n, 1, 0) if n&1 else b(n-1, 1, 0) if n else 0
print([a(n) for n in range(41)]) # Michael S. Branicky, Sep 29 2022

a(2*n-1) = a(2*n) for n >= 1. - David A. Corneth, Sep 29 2022

a(n) = A326027(n) - A357414(n). - Max Alekseyev, Mar 01 2025

a(24)-a(34) from Michael S. Branicky, Sep 29 2022
a(35)-a(70) from David A. Corneth, Sep 29 2022
a(0) prepended and terms a(71) onward added by Max Alekseyev, Mar 06 2025

A357416 Number of nonempty subsets of {1..n} whose elements have an even root mean square.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 8, 11, 13, 26, 46, 81, 169, 284, 482, 857, 1461, 2548, 4370, 7917, 14181, 25648, 47330, 87457, 163291, 302678, 568974, 1064393, 1993805, 3742588, 7030646, 13231519, 24871349, 46994382, 88657700, 167876827, 318263561, 604694212, 1150634498

Offset: 1

Ilya Gutkovskiy, Sep 27 2022

nonn

			a(9) = 8 subsets: {2}, {4}, {6}, {8}, {1, 3, 5, 8, 9}, {3, 4, 5, 7, 9}, {1, 3, 5, 6, 8, 9} and {3, 4, 5, 6, 7, 9}.

Max Alekseyev, Table of n, a(n) for n = 1..100

Cf. A339454, A357356, A357412, A357414, A357415.

a(n) = A339454(n) - A357415(n).

a(24)-a(41) from Alois P. Heinz, Sep 27 2022

A357411 Number of nonempty subsets of {1..n} whose elements have an odd harmonic mean.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 6, 6, 7, 9, 10, 10, 11, 13, 26, 26, 27, 45, 46, 74, 93, 99, 100, 162, 163, 165, 166, 458, 459, 865, 866, 866, 1647, 1669, 2724

Offset: 1

Ilya Gutkovskiy, Sep 27 2022

			a(11) = 10 subsets: {1}, {3}, {5}, {7}, {9}, {11}, {2, 6}, {2, 3, 6}, {3, 6, 10} and {3, 5, 6, 10}.

Cf. A339453, A357355, A357412, A357413, A357415.

from fractions import Fraction
from functools import lru_cache
def cond(s, c): h = c/s; return h.denominator == 1 and h.numerator&1
@lru_cache(maxsize=None)
def b(n, s, c):
    if n == 0: return int (c > 0 and cond(s, c))
    return b(n-1, s, c) + b(n-1, s+Fraction(1, n), c+1)
a = lambda n: b(n, 0, 0)
print([a(n) for n in range(1, 18)]) # Michael S. Branicky, Sep 29 2022

a(p) = a(p-1) + 1 for prime p > 2. - Michael S. Branicky, Sep 30 2022

a(24)-a(35) from Michael S. Branicky, Sep 30 2022

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A339454 Number of subsets of {1..n} whose root mean square is an integer.

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A357413 Number of nonempty subsets of {1..n} whose elements have an odd geometric mean.

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A357416 Number of nonempty subsets of {1..n} whose elements have an even root mean square.

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Extensions

A357411 Number of nonempty subsets of {1..n} whose elements have an odd harmonic mean.

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Examples

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Python

Formula

Extensions